An aspiration level adjustment rule is "realistic" if the aspiration level is (almost always) set to be an average of its previous value and the best average performance so far encountered. It is "ambitious" if the aspiration level is set to exceed the maximal average performance by some constant infinitely often. We analyze a case-based decision maker with a realistic-but-ambitious aspiration level adjustment rule facing a multi-armed bandit repeatedly. Though unaware of the payoff distributions corresponding to the arms of the bandit, the decision maker will asymptotically choose only expected-utility maximizing acts. Journal of Economic Literature Classification Numbers: C6, C61, D7, D72, D8, D81, D83.